## Abstract

Photonic biosensors offer label-free detection of biomolecules for applications ranging from clinical diagnosis to food quality monitoring. Both sensors based on Mach-Zehnder interferometers and ring resonators are widely used, but are usually read-out using different schemes, making a direct comparison of their fundamental limit of detection challenging. A coherent detection scheme, adapted from optical communication systems, has been recently shown to achieve excellent detection limits, using a simple fixed-wavelength source. Here we present, for the first time, a theoretical model to determine the fundamental limit of detection of such a coherent read-out system, for both interferometric and resonant sensors. Based on this analysis, we provide guidelines for sensor optimization in the presence of optical losses and show that interferometric sensors are preferable over resonant structures when the sensor size is not limited by the available sample volume.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Photonic biosensors enable label-free detection of trace amounts of biomolecules and real-time monitoring of bio-chemical reactions. Together with their compact size, which enables the integration of several sensors on the same chip to provide multiplexed operation, this makes these sensors well suited for applications in clinical diagnosis, environmental safety or food quality monitoring [1–6]. For most clinical applications it is critical to detect very low concentrations of the analyte in a given sample, of the order of a few ng/ml [7], which corresponds roughly to refractive index changes below 10^{−6} RIU [3]. The lowest concentrations that can be reliably detected is commonly referred to as the limit of detection (LOD), and depends both on the sensitivity of the sensor and the noise floor of the read-out system [8,9]. In the last years, a wide variety of sensors based on the shift of the resonance wavelength of rings and the power transmission of Mach-Zehnder interferometers has been studied [10–16], but only a few systematic studies on the fundamental bounds of the LOD are available. For resonant sensors, the use of a simple peak-search method to determine the resonance wavelength led to the conclusion that resonators with higher *Q*-factors yield sensors with lower (better) LOD [17–20]. Interferometric sensors interrogated with a white light source were studied in [21, 22], and it was shown that using a Fast-Fourier transformation technique highly accurate phase-read-out was possible. In two consecutive papers this technique was extended to resonant structures revealing that the best LOD is then obtained for intermediate *Q*-factors [23, 24]. Interferometric structures were shown to outperform resonant sensors by a thin margin when interrogated with such a broadband source [25].

However, the read-out techniques described so far require either a tunable source, or some type of spectrum analyzer, which increases the cost of the overall system. For point of care devices, in which cost and complexity should be kept low, a fixed, single wavelength interrogation scheme is preferable. Interferometric sensors are particularly amenable to this type of read-out [26–28], and most of their conventional limitations, such as sensitivity fading and directional ambiguity, have recently been overcome using advanced phase detection techniques [11, 12, 15, 29]. Indeed, some of the lowest limits of detection, of the order of 10^{−7} RIU, and better have been demonstrated with such systems [30]. Among the advanced phase detection schemes, the coherent technique used in [12, 15] is particularly straightforward, and builds on the solid theoretical foundations of coherent optical communication systems [31].

In this work, we examine, for the first time, the intrinsic limitations of both interferometric and resonant sensors with coherent phase read-out. We derive simple analytical expressions for their limit of detection that take into account electronic noise, relative intensity noise and phase noise in the system, as well as the inevitable losses due to the interaction with the aqueous solution that contains the analyte. We then outline clear design rules to optimize the limit of detection of these devices. We also show that, due to their simplicity, interferometric structures are preferable over resonant structures, unless the amount of available analyte limits the sensor size. We believe that these results also provide valuable insight for the optimization of sensors with other read-out techniques, particularly Mach-Zehnder interferometers with conventional read-out operated at the quadrature point.

The paper is organized as follows. In section 2 we establish a simple systemic view of photonic biosensors, providing precise definitions for sensitivity and limit of detection. We use these definitions to develop mathematical models for coherently detected Mach-Zehnder interferometers (section 3), ring resonators (section 4), and how they are affected by relative intensity and phase noise (section 5). Our sensor design guidelines are then presented in section 6, after which conclusions are drawn.

## 2. A systemic view of photonic biosensors

The cross-section of a typical photonic strip waveguide is shown in Fig. 1(a), consisting of the substrate, the guiding core, and the surrounding medium or cladding. At a given vacuum wavelength *λ*_{0} the first order mode propagates in *z*-direction with an effective refractive index *n*_{s}, and its evanescent field penetrates into the cladding layer. In a biosensor, the medium surrounding the core is normally an aqueous solution, and as its refractive index, *n _{c}*, changes, so does the effective index of the sensing waveguide,

*n*

_{s}. If the waveguide surface is functionalized, as illustrated in Fig. 1(b), the analyte will preferentially adhere to its surface, changing the mass density,

*ρ*, or thickness,

_{s}*h*, of the adhered layer, and consequently altering the effective mode index. To quantify this change the waveguid

*e*sensitivit

*y*is defined as:

*n*},

_{c}, h, ρ_{s}*S*

_{wg}has units of RIU/RIU, RIU/nm and RIU/(pg/mm

^{2}), respectively. In order to measure the resulting change in effective index, Δ

*n*

_{s}=

*S*

_{wg}ΔΓ, the waveguide is integrated in a sensing architecture, which is usually either based on an interferometer or a ring resonator. The conventional interferometric architecture is schematically illustrated in Fig. 2(a): light from a coherent laser source with power

*P*0 and a fixed wavelength

*λ*

_{0}is split among the sensing arm, which is exposed to analyte, and the reference arm, with the effective index

*n*

_{r}, which does not not interact with analyte; we assume both arms are of length

*L*. As light propagates through the arms it accumulates a phase difference

*ϕ*= 2

*π*(

*n*

_{s}−

*n*

_{r})

*L*/

*λ*

_{0}which is converted to the photocurrent

*i*at the output of the interferometer. The

*architectural sensitivity*is thus given by [9] and the overall sensitivity of the sensor is

The photocurrent *i* will, however, be contaminated by both the shot noise, with standard deviation *σ*_{shot}, and the equivalent noise of the electronics circuitry needed for amplification in the TIA (trans-impedance amplifier), with standard deviation *σ*_{TIA}. The total current noise can thus be considered to have a variance ${\sigma}^{2}={\sigma}_{\text{shot}}^{2}+{\sigma}_{\text{TIA}}^{2}$. The limit of detection of the system is then defined as a change in Γ that yields a signal three times larger than *σ* [17]:

## 3. Limit of detection of coherent interferometric sensors

For the conventional interferometer [19, 32, 33] shown in Fig. 2(a), the photo-current can be calculated as follows. The relation between the complex envelopes *a*_{0} and *a _{s}, a_{r}* at the input and output of the sensing and reference arms is given by:

*α*the electrical field attenuation of the sensing mode due to the interaction with the normally aqueous analyte, and ${a}_{0}=\sqrt{{P}_{0}/2}$. The waves

*a*and

_{s}*a*are recombined and photodetected with responsivity

_{r}*R*, yielding a photocurrent [34]:

Applying Eq. (2) we find that the architectural sensitivity is then given by

*n*

_{s}−

*n*

_{r})

*L*/

*λ*

_{0}=

*m*/2 + 1/4 with

*m*a natural number including zero. The well known limitations of sensitivity fading and directional ambiguity of interferometric sensors arise directly from the sinusoidal dependence of the architectural sensitivity. Note that the conventional interferometer could benefit from a signal enhancement by a factor of two if the output Y-junction is substituted by directional coupler and differential detection is employed.

Both limitations can be overcome with the coherent architecture, shown in Fig. 2(b). In this approach, an optical hybrid, typically consisting of a 2 × 4 or 2 × 3 multimode interference coupler, is used at the output of the interferometer, to recover the complex signal [31]:

^{∗}denotes complex conjugation and ∢ the phase. Note that the complex signal and the resultant signal to noise ratios are equivalent for the configurations based on a 2 × 3 and 2 × 4 multimode interference coupler [31]. As illustrated in Fig. 3, the detection process can then be thought of a movement of the current “vector” in the complex plane. It is apparent from this figure that the LOD will correspond to a movement in the complex plane |Δ

*i*| = |

_{c}*∂i*/

_{c}*∂n*

_{s}· Δ

*n*

_{s}| = 3

*σ*so that both measurements can be clearly distinguished. The rate of change of this vector is given by the architecture sensitivity [see Eq. (2)], which in this case becomes:

It is noteworthy that the architectural sensitivity of the coherent read-out scheme is the same as that of a conventional system operated at the quadrature point. Furthermore, as show in [31], the shot noise and the thermal TIA noise of the system can be transferred to the ‘I’ and ‘Q’ outputs as white Gaussian noise (see Fig. 3):

*q*is the charge of the electron,

*B*is the bandwidth of a low-pass filter, which is roughly the inverse of integration time of output signal, and

_{l}*η*

_{TIA}is the input-referred current noise density (in units of $\mathrm{A}/\sqrt{\text{Hz}}$) of each amplifier, which have been considered identical for both I and Q channels (see Fig. 2). In calculating the shot noise, we are assuming that the photodiode dark current can be neglected. It is worth noting that

*σ*=

*σ*=

_{I}*σ*because the noise that is orthogonal to the signal movement is irrelevant (see Fig. 3). Combining Eqs. (4), (9) and (10) then directly yields one of the main results of this paper, a closed expression for the LOD of a coherently detected interferometric sensor:

_{Q}To illustrate this, we consider the parameters shown in Table 1, which assume a conventional silicon-wire waveguide with TM polarization and a water cladding with power absorption coefficient 12 cm^{−1} (at *λ*_{0} = 1550 nm). Rigorous electromagnetic analysis of this structure show waveguide sensitivity and losses of *S*_{wg} ≈ 0.8, *α* ≈ 480 m^{−1}, respectively. Also, a value of thermal power density ${\eta}_{\text{TIA}}=3\text{pA}/\sqrt{\text{Hz}}$ (Thorlabs PDA10CS-EC) is considered. The LOD that results for such a system is shown in Fig. 4, where a clear optimum for the sensor length is observed around *αL ~* 1, i.e. length of only *L ~* 2mm. This shows that the interferometric sensing structure with coherent read-out, has the intrinsic capability to achieve excellent LOD values, with moderate lengths of the sensing arms which can be efficiently implemented in high contrast integrated optical platforms by means of compact spirals. While there are many other sources of noise, including amplitude and phase noise of the source, mechanical and thermal noise, which can degrade the attainable *LOD* in a practical set-up, we believe that establishing the fundamental limits of the LOD is a first step towards a more complete understanding that paves the way towards experimental results with improved LOD. From Fig. 4 we furthermore observe that the optimum length is quite similar if only shot noise or only electric noise is considered. The optimum length can be obtained analytically if we assume that system is limited either by shot noise (*σ*_{shot} ≫ *σ*_{TIA} – high input power) or electrical noise (*σ*_{TIA} ≫ *σ*_{shot} – low input power), as shown in Table 2.

## 4. Coherently detected ring

Resonant structures are often used in sensing applications, because they enable the realization of large effective interaction lengths with very compact structures. This advantage comes at the expense of some kind of tuning mechanism to ensure that the resonance condition is met. In this section we investigate the performance of a coherently detected ring resonator, as illustrated in Fig. 2(c). Note that the ring can be a simple circle or a more elaborate “folded” spiral. In either case, for a ring length *L*, the relation between the complex envelopes at the input and output of the sensing and reference arms is given by [35]:

*e*

^{−}

*are the round trip amplitude losses,*

^{αL}*ϕ*= 2

*πn*

_{s}

*L*/

*λ*

_{0}is the phase shift and we assume loss-less coupling between the ring and bus waveguide, i.e.

*t*

^{2}+

*κ*

^{2}= 1, with

*κ*and

*t*the coupling and transmission coefficient, respectively. Recalculating Eqs. (8) through (10), and after some algebra outlined in the appendix, we arrive at the following expression for the limit of detection:

We observe that the limit of detection generally depends on *ϕ* and hence on *n*s itself, as expected from the resonant nature of the structure. The best limit of the detection is obtained when the ring is tuned to its resonance, i.e. *ϕ* = *m* · 2*π* with *m* a natural number. This can be achieved by either heating the ring, or tuning the laser source; note that in the Mach-Zehnder no such tuning is required. Figure 5(a) shows the limit of detection as a function of the two remaining design parameters of the system: *t* and *αL*. Several revealing conclusions can be drawn from this figure. First, for rings with small radii, *αL* ≪ 1, which are often used in sensing, a good LOD is only achieved when the system is critically coupled. As the length of the ring is increased, the requirements on the coupling are significantly relaxed [see Fig. 5(b)], which improves fabrication tolerances because the coupling depends heavily on the small gap between the bus waveguide and the ring. Interestingly, the use of such long ring cavities with is equally beneficial when using ring-resonator with the conventional wavelength-swept read-out [36]. Second, given a certain *αL*, the optimum LOD is achieved for critical coupling, i.e. *t* = _{e}^{−}* ^{αL}*, shown with a blue curve in Fig. 5(a). If both resonance and critical coupling can be achieved simultaneously, the LOD is virtually constant for

*αL <*1 as illustrated in Fig. 5(c) -solid blue line. From this figure we also see that the best limit of detection is of the same order of magnitude as for the much simpler Mach-Zehnder interferometer, which corresponds to the

*t*= 0 case (all the power is coupled into the ring) -solid black line. Indeed, if the ring is critically coupled but not on resonance small rings with

*αL*≪ 1 are outperformed by Mach-Zehnder interferometers of the same length. For the limiting case

*αL <*1 with the ring tuned to resonance, and assuming critical coupling, Eq. (13) reduces to ${\text{LOD}}_{\text{RR}}=3\sqrt{2{B}_{l}}{\lambda}_{0}\alpha \sqrt{2{\eta}_{\text{TIA}}^{2}+qR{P}_{0}}/({P}_{0}R{S}_{\text{wg}}\pi )$, which for the cases of dominant shot or electrical noise can be further simplified to the expressions given in Table 2. Remarkably, with the exception of scaling factor of ~ 1.4 these expression are identical to those obtained for the much simpler Mach-Zehnder structure.

## 5. Effects of laser phase noise and RIN

In this section we will develop simple expressions for the effects of laser phase noise (section 5.1) and relative intensity noise (section 5.2) on the sensor output, which are then used to assess the impact on the LOD (section 5.3). It must be highlighted that, contrary to shot and thermal noises that are added at the output of the I and Q channels independently, laser induced noise will generate correlated noises in both channels. Thus instead of circle shaped Gaussian probability functions (as illustrated in Fig. 3), laser induced noise will cause more complicated probability distributions in the complex plane.

#### 5.1. Effect of laser phase noise on complex photocurrent

Following [37] we consider a laser phase noise described by a ${a}_{0}(t)=\sqrt{{P}_{0}/2}\cdot \mathrm{exp}(j\psi (t))$ with Lorentzian lineshape of full width half maximum (FWHM) linewidth Δ*ν*. For most good quality lasers Δ*ν* can be considered to be below the 100 MHz range: a frequency span in which most optical components show an almost constant amplitude response, and an almost linear phase response. In other words, to study the effect of phase noise the frequency response of the reference or sensing arms can be approximated as

*ω*= 2

*πc*/

*λ*is the angular frequency,

*c*≈ 3 · 10

^{8}m/s the speed of light in vacuum, and

*τ*= −

_{s,r}*d*∢

*H*(

_{s,r}*ω*

_{0})/

*dω*is the group delay. Under this approximation, light waves at the coherent receiver input can be written as

Therefore, the complex photocurrent at the receiver output becomes

*ψ*(

*t*) =

*ψ*(

*t*−

*τ*)−

_{s}*ψ*(

*t*−

*τ*). This expression clearly shows that laser phase noise maps into output phase noise, i.e. the noisy complex signal

_{r}*i*(

_{c}*t*) describes a circle arc in the complex plane. Furthermore, as the phase difference spectral density is a simple function [37], the phase error standard deviation

*σ*

_{Δ}

*can be approximated as where Δ*

_{ψ}*τ*= |

*τ*

_{s}−

*τ*

_{r}| is the group delay difference and

*B*is the measurement bandwidth which, for a biosensing setup, can be considered to fulfill

_{l}*B*≪ 1/Δ

_{l}*τ*(typically

*B*100 kHz). Thus the standard deviation of the photocurrent arc becomes

_{l}<*H*(

_{s}*ω*)|, |

_{o}*H*(

_{r}*ω*)|, and Δ

_{o}*τ*have been included: where

*n*and

_{g,s}*n*are the group indices of the sensing and reference signal, respectively. Table 3 and Eq. (18) enable us to easily evaluate the impact of phase noise in both types of systems.

_{g,r}#### 5.2. Effect of laser RIN on complex photocurrent

We will consider RIN described by ${a}_{0}(t)=\sqrt{{P}_{0}/2}(1+1/2\cdot r(t))$, where *r*(*t*) exhibits a power spectral density *S _{rr}* (

*f*) which can be approximated as a white noise

*S*(

_{rr}*f*) =

*S*with values in the range of −110 to −160 dB/Hz [38] [see appendix (A.2) for a description of the noise model]. Under the low noise approximation,

_{RIN}*r*(

*t*) ≪ 1, it can be shown that the influence of RIN on the output photocurrent can be modeled through a linear filtering operation [see appendix (A.2)]. Thus, the calculation of the spectral power density of the output photocurrent in the narrow measurement bandwidth

*B*(typically well below 100 kHz) only requires modeling the sensing and reference arms’ frequency response in that reduced frequency span. We can therefore use the same narrow-band approximation as in the previous section [see Eq. (14)]. In doing so, the waves at the input of the coherent receiver become:

_{l}This expression clearly shows that RIN maps into output amplitude noise. Furthermore, as the delayed replicas of *r*(*t*) can be considered uncorrelated, the standard deviation of the RIN induced output photocurrent becomes.

#### 5.3. Impact of RIN and phase noise on the sensor LOD

In the previous subsections we have shown that laser RIN and phase noise map into amplitude and phase noise of the complex output photocurrent, respectively. This situation is illustrated for the interferometric case in Fig. 6(a) and for the resonant case in Fig. 6(b). The path that the complex photocurrent will follow as a consequence of a variation of the refractive index of the sensing arm is plotted with a continuous blue line. This movement will follow a circle in the interferometric case [Fig. 6(a)], and a resonant loop in the resonant case [Fig. 6(b)]. For the interferometric case it is clear that only the phase noise (*σ _{PN}*) will cause a degradation of the LOD as the RIN (

*σ*) is always orthogonal to complex current movement due to sensing and will be eliminated in the phase recovery stage. This is true for any position on the circle, so that interferometric sensors always reject RIN without any tuning. For the resonant case the same is true if the system is exactly at resonance, where only phase noise will cause LOD degradation because RIN will also be orthogonal to the complex current movement due to sensing. However, when moving away from resonance the effects of RIN become noticeable.

_{RIN}To achieve the intrinsic limit of detection stated in sections 3 and 4 phase noise should be kept well below the TIA and shot noises. As TIA noise exceed shot noise for the case under study (see Table 1), a reasonable condition to achieve the intrinsic limit of detection will be to set ${\sigma}_{PN}^{2}<{\sigma}_{TIA}^{2}/10$. Equation (18) and Table 3 have been used to calculate the maximum allowable laser linewidth for which ${\sigma}_{PN}^{2}={\sigma}_{TIA}^{2}/10$ given the system parameters in Table 1. For the interferometric case we have set the length to its optimum value *L* = 1/*α*, and we have introduced a reasonable amount of group index difference (10 %) between sensing and reference arms: *n _{g,r}* = 3,

*n*= 3.3. For the resonant case we have set an resonant enhancement of 1/

_{g,s}*αL*= 10, and we have chosen an almost critical coupling condition

*t*= 0.98 · exp (−

*αL*) (corresponding to an extinction at resonance of ~ 20dB). Maximum laser linewidth values have been found to be Δ

*ν*= 1.9 GHz for the interferometric sensor and Δ

*ν*= 11 MHz for the resonant one.

These results show that interferometric sensors with coherent read-out are relatively insensitive to laser phase noise, so they are good candidates for low cost sensors as those required in point of care applications. On the contrary, resonant sensors with coherent reading are quite sensitive to laser phase noise. It has also been shown that both systems are able to cancel out the RIN noise which could be advantageous as compared with non-coherent reading systems.

## 6. General sensor design guidelines

From the expressions in Table 2 and Figs. 4 and 5, and the results of section 5 several conclusions about the coherently detected sensing system can be drawn. We believe that these conclusions provide clarification and useful insight for other sensor architectures, too:

- The LOD improves by reducing the losses due to water absorption,
*α*, and the wavelength,*λ*_{0}. Both factors favor sensors operating in the visible, where the water absorption is much lower than in the infrared. - An enhanced waveguide sensitivity,
*S*_{wg}, only improves the LOD if it does not result in additional losses of the waveguide mode. If the losses of the waveguide mode arise mostly from the overlap of the mode with the lossy aqueous solution, i.e.*α*=*α*_{water}*S*_{wg}, the LOD is independent on waveguide sensitivity. - For Mach-Zehnder based sensors, the optimum arm length is roughly
*L*≈ 1/*α*, and no tuning is required. If*α*≈*α*_{water}*S*_{wg}, an enhanced waveguide sensitivity thus results in shorter sensors. - Ring-resonator based sensors do not offer a significant enhancement in LOD compared to Mach-Zehnder based sensors but require wavelength tuning which results in a more complex read-out system. Interferometric architecture seems preferable whenever it is possible to allocate spirals with the optimum length
*L*≈ 1/*α*on the chip. - If very compact sensor are required, rings with the largest acceptable value of
*L*should be chosen (e.g. by using a “folded” spiral), as this will relax the requirements on the coupling and thus facilitate device fabrication and read-out. - Coherently detected interferometric sensors can be designed so that they are almost unaffected by laser RIN and phase noise. This makes this type of sensors good candidates for applications where the cost of the reading systems is important.
- Coherently read-out ring resonator based sensors are sensitive to laser phase noise and they are prone to exhibit greater sensitivity to wavelength drifts.

## 7. Conclusions

Coherently detected biosensor systems have the potential to achieve excellent limits of detection using a simple, fixed-wavelength read-out system. We have analyzed such systems in detail, providing closed-form expressions for their fundamental limits when used both with Mach-Zehnder interferometers and ring resonators. Our results provide a clear comparison of the interferometric and resonant sensors, as well as specific guidelines to improve the limit of detection of the systems, which we believe are of great interest for the development of high performance biosensing systems.

## A. Appendix

## A.1. Coherent resonant phase read-out -LOD derivation

The derivation of Eq. (13) is outlined in the following. We start by calculating the complex current (see Eq. (8)) with the ring transfer function defined in Eq. (12), which yields

To obtain the architectural sensitivity given by Eq. (2), we take the derivative of the complex current with respect to *n*s using the chain rule *∂i*_{c}/*∂n*_{s} = (*∂i*_{c}/*∂ϕ*)(*∂ϕ*/*∂n*_{s}) = (*∂i*_{c}/*∂ϕ*)(2*πL*/*λ*_{0}):

On the other hand, from Eq. (10) the shot noise is

## A.2. Relative intensity noise -derivation

Here we show that the effect of the relative intensity noise on the output current can be modeled as a linear filtering operation. We incorporate RIN by setting

where*r*(

*t*) accounts for RIN. Note that the expression of the RIN noise matches with the standard definition [39] under the small RIN approximation (

*r*

^{2}(

*t*) ≪ 1),

*r*(

*t*) is the normalized power fluctuation

*r*(

*t*) =

*δP*(

*t*)/

*P*

_{0}whose power spectral density is

*S*(

_{rr}*f*). The same model as in Eq. (25) is assumed for the input waves to the coherent receiver, with filtered versions of the laser signal

*a*(

_{s,r}*t*) =

*a*(

_{o}*t*)∗

*h*(

_{s,r}*t*) ([]∗[] represents the convolution):

Using Eq. (8), the output photocurrent can then be calculated as

*r*(

_{s}*t*)

*r*(

_{r}*t*) has been neglected under the small RIN approximation. Remarkably this expression shows that, laser RIN generated at the input of the systems in Figs. 2(b) and 2(c) can be transferred to their output by linear time invariant operations (filtering). This implies that the output photocurrent power spectral density inside the measurement bandwidth

*B*only depends of the frequency response of the reference and sensing arms in that reduced frequency range. As usually the measurement bandwidth is very small (

_{l}*B*is typically below 100kHz) this allows us to approximate the frequency response of reference and sensing arms through Eq. (14) even if RIN noise is spectrally broad.

_{l}## Funding

Ministerio de Economía y Competitividad (TEC2016-80718-R); Marie Skłodowska-Curie (713721).

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